Point searching in real singular complete intersection varieties – algorithms of intrinsic complexity 1
Bernd Bank
2, Marc Giusti
3, Joos Heintz
4July 21, 2011
Abstract
Let X1, . . . , Xn be indeterminates over Q and let X := (X1, . . . , Xn) . Let F1, . . . , Fp be a regular sequence of polynomials in Q[X] of degree at most d such that for each 1≤k≤p the ideal (F1, . . . , Fk) is radical. Suppose that the variables X1, . . . , Xn
are in generic position with respect to F1, . . . , Fp. Further suppose that the poly- nomials are given by an essentially division-free circuit β in Q[X] of size L and non-scalar depth `.
We present a family of algorithms Πi and invariants δi of F1, . . . , Fp, 1 ≤ i ≤ n−p, such that Πi produces on input β a smooth algebraic sample point for each connected component of {x ∈Rn |F1(x) =· · ·= Fp(x) = 0} where the Jacobian of F1= 0, . . . , Fp= 0 has generically rank p.
The sequential complexity of Πi is of order L(n d)O(1)(min{(n d)c n, δi})2 and its non-scalar parallel complexity is of order O(n(`+ logn d) logδi) . Here c > 0 is a suitable universal constant. Thus, the complexity of Πi meets the already known worst case bounds. The particular feature of Πi is its pseudo-polynomial and intrinsic complexity character and this entails the best runtime behavior one can hope for. The algorithm Πi works in the non-uniform deterministic as well as in the uniform probabilistic complexity model. We exhibit also a worst case estimate of order (nnd)O(n) for the invariant δi.
Keywords: real polynomial equation solving, intrinsic complexity, singularities, polar, copolar and bipolar variety, degree of variety
MSC: 68W30, 14P05, 14B05, 14B07, 68W10
1Research partially supported by the following Argentinian, French and Spanish grants: CONICET PIP 2461/01, UBACYT X-098, PICT–2010–0525, Digiteo DIM 2009–36HD, MTM2010-16051.
2Humboldt-Universit¨at zu Berlin, Institut f¨ur Mathematik, 10099 Berlin, Germany.
bank@math.hu-berlin.de
3CNRS, ´Ecole Polytechnique, Lab. LIX, 91228 Palaiseau Cedex, France.
marc.giusti@polytechnique.fr
4Departamento de Computaci´on, Universidad de Buenos Aires and CONICET, Ciudad Univ., Pab.I, 1428 Buenos Aires, Argentina, and Departamento de Matem´aticas, Estad´ıstica y Computaci´on, Facultad de Ciencias, Universidad de Cantabria, 39071 Santander, Spain.
joos@dc.uba.ar & joos.heintz@unican.es
1 Introduction
Before we start to explain the main results of this article and their motivations, we introduce some basic notions and notations.
Let Q, R and C be the fields of rational, real and complex numbers, respectively, let X := (X1, . . . , Xn) be a vector of indeterminates over C and let F1, . . . , Fp be a regular sequence of polynomials in Q[X] defining a closed, Q–definable subvariety S of the n– dimensional complex affine space An := Cn. Thus S is a non–empty equidimensional affine variety of dimension n−p, i.e., each irreducible component of S is of dimension n−p. Said otherwise, S is a closed subvariety of An of pure codimension p (in An).
Let AnR := Rn be the n–dimensional real affine space. We denote by SR := S ∩AnR the real trace of the complex variety S. Moreover, we denote by Pn the n–dimensional complex projective space and by PnR its real counterpart. We shall use also the following notations:
{F1 = 0, . . . , Fp = 0}:=S and {F1 = 0, . . . , Fp = 0}R :=SR.
We call the regular sequence F1, . . . , Fp reduced if the ideal (F1, . . . , Fp) generated in Q[X] is the ideal of definition of the affine variety S, i.e., if (F1, . . . , Fp) is radical. We call (F1, . . . , Fp) strongly reduced if for any index 1 ≤ k ≤ p the ideal (F1, . . . , Fk) is radical. Thus, a strongly reduced regular sequence is always reduced.
A point x of An is called (F1, . . . , Fp)–regularif the Jacobian J(F1, . . . , Fp) :=h∂F
j
∂Xk
i
1≤j≤p 1≤k≤n
has maximal rank p at x. Observe, that for each reduced regular sequence F1, . . . , Fp defining the variety S, the locus of (F1, . . . , Fp) –regular points of S is the same. In this case we call an (F1, . . . , Fp) –regular point of S simply regular(or smooth) or we say that S is regular (or smooth) at x. The set Sreg of regular points of S is called the regular locus, whereas Ssing := S\Sreg is called the singular locus of S. Remark that Sreg is a non–empty open and Ssing a proper closed subvariety of S. We say that a connected component C of SR is generically smooth if C contains a smooth point.
We suppose now that there are natural numbers d, L and ` and an essentially division–
free arithmetic circuit β in Q[X] with p output nodes such that the following conditions are satisfied.
- The degrees degF1, . . . ,degFp of the polynomials F1, . . . , Fp are bounded by d. - The p output nodes of the arithmetic circuit β represent the polynomials
F1, . . . , Fp by evaluation.
- The size and the non–scalar depth of the arithmetic circuit β are bounded by L and `, respectively.
For the terminology and basic facts concerning arithmetic circuits we refer to [22, 13, 11].
Suppose that the variables X1, . . . , Xn are in generic position with respect to the variety S. Observe that we allow SR to have singular points.
In this paper we design for each 1 ≤ i≤ n−p a non–uniform deterministic or uniform probabilistic procedure Πi and an invariant δi satisfying the following specification.
(i) The invariant δi is a positive integer depending on F1, . . . , Fp and having asymptotic order not exceeding (nnd)O(n). We call δi the degree of the real interpretation of the equation system F1 = 0, . . . , Fp = 0 .
(ii) The algorithm Πi decides on input β whether the variety S contains a smooth real point and, if it is the case, produces for each generically smooth connected component of S a suitably encoded real algebraic sample point.
(iii) In order to achieve this goal, the algorithm Πi performs on input β a computation in Q with L(n d)O(1)(min{(n d)c n, δi})2 arithmetic operations (additions, subtractions, multiplications and divisions) which become organized in non–scalar depth O(n(`+ logn d) logδi) with respect to the parameters of the arithmetic circuit β (here c >0 is a suitable universal constant).
This is the outcome of our main result, namely Theorem 14 below and the three remarks following the theorem.
Although we were not able to derive a better worst case bound as (nnd)O(n) for the invariant δi (see Proposition 8, 12 and Observation 11 below) the worst case complexity of the procedure Πi meets the already known extrinsic bound of (n d)O(n) for the elimination problem under consideration (compare the original papers [24, 12, 34, 28, 29, 29, 30, 35, 9]
and the comprehensive book [10]).
The complexity of the procedure Πi depends polynomially on the extrinsic parameters L, `, n and d and on the degree δi of the real interpretation of the equation system F1 = 0, . . . , Fp = 0 which represents an intrinsic parameter measuring the input size of our computational task. In this sense we say that the procedure Πi is of intrinsic complexity.
Since the complexity L(n d)O(1)(min{(n d)c n, δi})2 is polynomial in all its parameters, including the intrinsic parameter δi we say that the procedure Πi ispseudo-polynomial. In fact, intrinsic complexity and pseudo-polynomiality constitute the best runtime behavior of Πi one can hope for.
In the case that SR is smooth and F1, . . . , Fp is a strongly reduced regular sequence in Q[X] there exist already pseudo-polynomial algorithms of intrinsic complexity which solve the computational task of item(ii) above (see [1, 3, 4]). The same is true for the singular hypersurface case, namely p := 1 , where {F1 = 0}R contains possibly singular points (see [8, 6, 7]). The methods developed in [1, 2, 4] cannot be applied directly when SR is singular. To overcome this difficulty we consider in Section 3.1 two families of smooth incidence varieties which parametrize the so-called copolar varieties of S introduced in Section 3.2.
For a given full rank matrix b ∈ A(n−i)×n the corresponding copolar variety of S is the Zariski closure of the set of all points x of S such that there exist p rows of b which generate the same linear space as the rows of the Jacobian of the equation system F1 = 0, . . . , Fp = 0 at x.
The procedure Πi is based on a geometrical and computational analysis of the dual polar varieties of the two families of incidence varieties (see [3, 4, 5] for the notion of a dual polar variety). These geometric objects are called bipolar varieties of S. They become introduced in Section 4.1. Important for the worst case complexity of the procedure are the degree estimates for the bipolar varieties developed in Section 4.2.
2 Preliminaries about polar varieties
Let notations be as in the Introduction. Let F1, . . . , Fp ∈ Q[X] be a reduced regular sequence defining a (non–empty) subvariety S of An of pure codimension p.
Let 1 ≤ i ≤ n − p and let a := [ak,l]1≤k≤n−p−i+1 0≤l≤n
be a complex ((n − p− i + 1) × (n+ 1) –matrix and suppose that a∗ := [ak,l]1≤k≤n−p−i+1
1≤l≤n has maximal rank n −p−i+
1 . In case (a1,0, . . . , an−p−i+1,0) = 0 we denote by K(a) := Kn−p−i(a) and in case (a1,0, . . . , an−p−i+1,0) 6= 0 by K(a) := Kn−p−i(a) the (n − p −i) –dimensional linear subvarieties of the projective space Pn which for 1≤ k ≤n−p−i+ 1 are spanned by the the points (ak,0 : ak,1 : · · · : ak,n) . In the first case we shall also use the notations K(a∗) and Kn−p−i(a∗) instead of K(a) and Kn−p−i(a) .
The classic and the dual ith polar varieties of S associated with the linear varieties K(a) and K(a) are defined as the closures of the loci of the (F1, . . . , Fp) –regular points of S where all (n−i+ 1) –minors of the respective polynomial ((n−i+ 1)×n) –matrix
∂F1
∂X1
· · · ∂F1
∂Xn
.. .
.. .
.. .
∂Fp
∂X1
· · · ∂Fp
∂Xn
a1,1 · · · a1,n
.. .
.. .
.. . an−p−i+1,1 · · · an−p−i+1,n
and
∂F1
∂X1
· · · ∂F1
∂Xn
.. .
.. .
.. .
∂Fp
∂X1
· · · ∂Fp
∂Xn
a1,1−a1,0X1 · · · a1,n−a1,0Xn
.. .
.. .
.. .
an−p−i+1,1−an−p−i+1,0X1 · · · an−p−i+1,n−an−p−i+1,0Xn
vanish. We denote these polar varieties bya is a real ((n−p−i+ 1)×(n+ 1) –matrix, we denote by
WK(a)(SR) :=WKn−p−i(a)(SR) :=WK(a)(S)∩AnR and
WK(a)(SR) :=WKn−p−i(a)(SR) :=WK(a)(S)∩AnR the real traces of WK(a)(S) and WK(a)(S) .
Observe that this definition of classic and dual polar varieties may be extended to the case that there is given a Zariski open subset O of An such that the equations F1 = 0, . . . , Fp = 0 intersect transversally at any of their common solutions in O and that S is now the locally closed subvariety of An given by
S :={F1 = 0, . . . , Fp = 0} ∩O, which is supposed to be non-empty.
In Section 4 we shall need this extended definition of polar varieties in order to establish the notion of a bipolar variety of a given hypersurface. For the moment let us suppose again that S is the closed subvariety of An defined by the reduced regular sequence F1, . . . , Fp.
In [3] and [4] we have introduced the notion of dual polar varieties of S (and SR) and motivated by geometric arguments the calculatory definition of these objects. Moreover, we have shown that, for a complex ((n−p−i+ 1)×(n+ 1)) –matrix a= [ak,l]1≤k≤n−p−i+1
0≤l≤n
with [ak,l]1≤k≤n−p−i+1
1≤l≤n generic, the polar varieties WK(a)(S) and WK(a)(S) are either empty or of pure codimension i in S. Further, we have shown that WK(a)(S) and WK(a)(S) are normal and Cohen–Macaulay (but non necessarily smooth) at any of their (F1, . . . , Fp) – regular points (see [5], Corollary 2 and Section 3.1). This motivates the consideration of the so–called generic polar varieties WK(a)(S) and WK(a)(S) , associated with complex ((n−p−i+ 1)×(n+ 1)) –matrices a which are generic in the above sense, as invariants of the complex variety S (independently of the given equation system F1 = 0, . . . , Fp = 0 ).
However, when a generic ((n−p−i+ 1)×(n+ 1)) –matrix a is real, we cannot consider WK(a)(SR) and WK(a)(SR) as invariants of the real variety SR, since for suitable real generic ((n −p−i+ 1) ×(n+ 1)) –matrices these polar varieties may turn out to be empty, whereas for other real generic matrices they may contain points (see [5], Theorem 1 and Corollary 2 and [8], Theorem 8 and Corollary 9).
For our use of the word “generic” we refer to [5], Definition 1.
In case that SR is smooth and a is real ((n−p−i+ 1)×(n+ 1)) –matrix, the real dual polar variety WK(a)(SR) contains at least one point of each connected component of SR, whereas the classic (complex or real) polar varieties WK(a)(S) and WK(a)(SR) may be empty (see [3] and [4], Proposition 2).
Polar varieties have a long story in algebraic geometry which goes back to Severi [38]
and Todd [42, 43] in the 1930’s. Originally they were used to establish numerical formu- las in order to classify singular algebraic varieties by their intrinsic geometric character or to formulate a manageable local equisingularity criterion which implies the Whitney conditions for analytic varieties. About 10 years ago they became also a fundamental tool for the design of pseudo-polynomial computer procedures with intrinsic complexity bounds which find for a given complete intersection variety S with a smooth real trace SR algebraic sample points for each connected component of SR. For details we refer to [41] and [5].
3 Copolar incidence varieties
3.1 Two families of copolar incidence varieties
Let d, n, p and i be natural numbers with 1 ≤ p ≤ n and 1 ≤ i ≤ n −p and let X := (X1, . . . , Xn) , B := [Bk,l]1≤k≤n−i
1≤l≤n , Λ := [Λr,s]1≤r,s≤p and Θ := [Θk,r]1≤k≤n−i
1≤r≤p be
matrices of indeterminates over C.
We fix for the rest of the paper a strongly reduced regular sequence F1, . . . , Fp ∈ Q[X] . Let d := max{degFs | 1 ≤ s ≤ p}, where degFs denotes the degree of the polynomial Fs. We denote by F := (F1, . . . , Fp) the sequence of these polynomials and by J(F) :=
h∂Fs
∂Xl
i
1≤s≤p 1≤l≤n
the Jacobian of F. Observe that the rank of J(F) is generically p on any irreducible component of the complex variety S :={F1 =· · ·=Fp = 0}. We write J(F)T for the transposed matrix of J(F) and for any point x ∈ An we denote by rk J(F)(x) the rank of the complex matrix J(F)(x) .
We are now going to introduce two families of varieties which we shall callcopolar incidence varieties. In order to define the first one we consider in the ambient space
Ti :=An×A(n−i)×n×Ap×p×A(n−i)×p the Q–definable locally closed incidence variety
Hi :={(x, b, λ, ϑ)∈Ti| x∈S, rkb =n−i, rkϑ=p, J(F)(x)Tλ+bTϑ= 0}.
Observe that the isomorphy class of Hi does not depend on the choice of the generators F1, . . . , Fp of the vanishing ideal of S. The canonical projection of Ti onto An maps Hi into S.
We are now going to state and prove three facts, namely Lemma 1 and Propositions 2 and 3 below, which will be fundamental in the sequel.
Lemma 1
Let (x, b, λ, ϑ) be a point of Hi. Then x belongs to Sreg and λ is a regular complex (p×p)–matrix. Moreover, the canonical projection of Ti onto An maps Hi onto Sreg and (Hi)R onto (SR)reg .
Proof
Let (x, b, λ, ϑ) be a point of Hi. Then b and ϑ are complex full rank matrices of size (n−i)×n and (n−i)×p, respectively. Therefore bTϑ is a complex full rank matrix of size n×p. From J(F)(x)Tλ+bTϑ = 0 we deduce that the complex (n×p) –matrix J(F)(x)Tλ and the matrix λ have rank p. This implies that the rank of J(F)(x) is p. Since x belongs to S we conclude that S is smooth at x. Thus we have x ∈ Sreg and λ is a regular complex (p×p) –matrix. By the way we have shown that the canonical projection of Ti onto An maps Hi into Sreg .
Consider now an arbitrary point x∈Sreg . Without loss of generality we may assume that the first p columns of J(F)(x) are C–linearly independent. Let λ be the (p×p) –identity
matrix Ip. Furthermore, let b :=
−J(F)(x) O(n−p−i)×p I(n−i−p)×n
and ϑ :=
Ip O(n−p−i)×p
,
where O(n−p−i)×p denotes the ((n−p−i)×p) –zero matrix.
Then b and ϑ are full rank matrices which satisfy the condition J(F)(x)Tλ+bTϑ= 0 . Since x belongs to S, we conclude that (x, b, λ, ϑ) is an element of Hi which be- comes mapped onto x under the canonical projection of Ti onto An. In particular, if x∈(SR)reg then λ, b and ϑ are real matrices and (x, b, λ, ϑ) belongs to (Hi)R. This implies that the canonical projection of Ti onto An maps Hi onto Sreg and (Hi)R onto
(SR)reg . 2
Proposition 2
Let Di be the closed subvariety of Ti defined by the conditions rkB < n−i or rkΘ< p. Then the polynomial equations
F1(X) = · · ·=Fp(X) = 0, X
1≤s≤p
Λr,s ∂Fs
∂Xl(X) + X
1≤k≤n−i
Bk,lΘk,r = 0, 1≤r ≤p, 1≤l≤n,
(1)
intersect transversally at any of their common solutions in Ti \ Di. Moreover, Hi is exactly the set of solutions of the polynomial equation system (1) outside of the locus Di.
In particular, Hi is an equidimensional algebraic variety which is smooth and of dimension n(n−i+ 1) +p(p−i−1)≥0.
Proof
One sees immediately that a point (x, b, λ, ϑ) ∈ Ti belongs to Hi if and only if it is a solution of the the polynomial equation system (1) outside of the locus Di. The Jacobian of the system (1) at such a point (x, b, λ, ϑ) has the following form
Lx :=
J(F)(x) Op×p · · · Op×p Op×(n−i) · · · Op×(n−i) Op×n(n−i)
∗
J(F)(x)T · · · On×p
..
. . .. ... On×p · · · J(F)(x)T
bT · · · On×(n−i)
..
. . .. ...
On×(n−i) · · · bT
D(1) .. .
D(p)
,
where D(r), 1≤r≤p, is the complex (n×n(n−i)) –matrix
D(r):=
ϑ1,r · · · ϑn−i,r · · · O1×(n−i)
..
. . .. ...
O1×(n−i) · · · ϑ1,r · · · ϑn−i,r
.
From Lemma 1 we conclude that the (p×n) –matrix J(F)(x) has maximal rank p. Since the matrix ϑ has full rank we infer that the (np×n(n−i)) –submatrix of Lx built up by D(1), . . . , D(r) has rank np. This implies that the Jacobian Lx has full rank. Therefore, the (n + 1)p equations of the system (1) intersect transversally at (x, b, λ, ϑ) and the algebraic variety Hi is smooth and of dimension
n+ (n−i)n+p2+ (n−i)p−(n+ 1)p=n(n−i+ 1) +p(p−i+ 1)
at this point. Thus Hi is an equidimensional variety which is empty or smooth and of dimension n(n −i+ 1) +p(p−i+ 1) (observe that 1 ≤ l ≤ n, 1 ≤ r ≤ p imply n(n−i+ 1) +p(p−i+ 1)≥0 ). From Lemma 1 we deduce that Hi is not empty. 2 For algorithmic applications Proposition 2 contains too many open conditions, namely the conditions rk B =n−i and rk Θ =p. By means of a suitable specialization of the matrices B and Θ we are going to eliminate these open conditions. However, we have to take care that these specialization process does not exclude to many smooth points of the variety S. The following result, namely Proposition 3 below seems to represent a fair compromise. We shall need it later for the task of finding smooth points of S. For the formulation of this proposition we need some notations.
Let B and Θ be the following matrices
B :=
B1,n−i+1 · · · B1,n ... · · · ... Bp,n−i+1 · · · Bp,n
and Θ:=
Θp+1,1 · · · Θp+1,p ... · · · ... Θn−i,1 · · · Θn−i,p,
.
Let σ be a permutation of the set {1, . . . , n} (in symbols, σ ∈Sym (n) ) and apply σ to the columns of the ((n−i)×n) –matrix
Ip Op×(n−p−i) B
O(n−p−i)×p In−p−i O(n−p−1)×(n−i)
.
In this way we obtain a ((n−i)×n) –matrix which we denote by Bi,σ. Furthermore, let Θi :=
Ip Θ
and ∆σ := deth∂F
s
∂Xσ(r)
i
1≤s,r≤p.
If we specialize in Bi,σ the submatrix B to b ∈ Ap×iand in Θi the submatix Θ to ϑ ∈ A(n−p−i)×p then the resulting complex matrices become denoted by bi,σ and ϑi, respectively.
We consider now in the ambient space
Fi :=An×Ap×i×Ap×p×A(n−p−i)×p a copolar incidence variety of more restricted type, namely
Hi,σ :={(x, b, λ, ϑ)∈Fi |x∈S, J(F)(x)Tλ+bTi,σϑi = 0}.
Observe that Hi,σ is a Q–definable locally closed subvariety of Fi whose isomorphy class does not depend on the choice of the polynomials F1, . . . , Fp of the vanishing ideal of S. In the statement of the next result we make use of the Kronecker symbol δr,l, 1≤r, l≤p which is defined by δr,l := 0 for r6=l and δr,r:= 1 .
Proposition 3
Let notations and definitions be as before. For the sake of simplicity assume that σ is the identity permutation of Sym(n). Then the polynomial equations
X
1≤s≤p
Λr,s ∂Fs
∂Xl(X) +δr,l= 0, 1≤r, l≤p, X
1≤s≤p
Λr,s ∂Fs
∂Xl
(X) + Θl,r = 0, 1≤r ≤p, p < l≤n−i, X
1≤s≤p
Λr,s ∂Fs
∂Xl(X) +Br,l = 0, 1≤r≤p, n−i < l≤n (2)
intersect transversally at any of their common solutions in Fi. Moreover, Hi,σ is exactly the set of solutions of the equation system (2). In particular, Hi,σ is a closed equidimen- sional algebraic variety which is empty or smooth and of dimension n−p.
The image of Hi,σ under the canonical projection of Fi onto An is the set of (smooth) points of S where ∆σ does not vanish. For each real point x∈S with ∆σ(x)6= 0 there exists a real point (x, b, λ, ϑ) of Hi,σ.
Proof
From the matrix identities
Bi,σTΘi =
Ip Op×(n−p−i)
O(n−p−i)×p In−p−i
BT Oi×(n−p−i)
Ip
Θ
=
Ip
Θ BT
=
1 · · · 0 ... · · · ... 0 · · · 1 Θp+1,1 · · · Θp+1,p
... · · · ... Θn−i,1 · · · Θn−i,p
B1,n−i+1 · · · Bp,n−i+1
... · · · ... B1,n · · · Bp,n
.
one deduces easily that a point (x, b, λ, ϑ) of Fi belongs to Hi,σ if and only if it is a solution of the polynomial equation system (2).
Let (x, b, λ, ϑ) be such a point of Hi,σ. Then the system (2) implies ∆σ(x)6= 0 and its
Jacobian may be organized as the matrix
L(i,σ)x :=
J(F)(x) Op×p · · · Op×p Op×(n−p) · · · Op×(n−p)
∗
J(F)(x)T · · · On×p ... . .. ... On×p · · · J(F)(x)T
Z · · · On×(n−p)
... . .. ...
On×(n−p) · · · Z
,
with
Z :=
Op×(n−p−i) Op×i
In−p−i O(n−p−i)×i
Oi×(n−p−i) Ii
.
From ∆σ(x)6= 0 we deduce that J(F)(x) has rank p. Thus L(i,σ)x has full rank. There- fore, the (n+ 1)p equations of the system (2) intersect transversally at (x, b, λ, ϑ) and the algebraic variety Hi,σ is smooth and of dimension n+pi+p2+(n−p−i)p−(n+1)p=n−p at this point. Thus Hi,σ is an equidimensional variety which is empty or smooth of di- mension n−p. For any point (x, b, λ, ϑ) of Hi,σ we have ∆σ(x)6= 0 and, in particular, S is smooth at x.
On the other hand, for x∈S with ∆σ(x)6= 0 we may consider λ:=∂Fs
∂Xl
−1
1≤s,l≤p, ϑ :=−∂Fs
∂Xl
T
1≤s≤p p<l≤n−i
·λ and b:=−∂Fs
∂Xl
T
1≤s≤p n−i≤l≤n
·λ.
Then the corresponding point (x, b, λ, ϑ) belongs to Hi,σ. Moreover, for x real we have that b, λ, ϑare also real and therefore (x, b, λ, ϑ) is a real point of Hi,σ. 2 In the sequel we shall refer to Hi and Hi,σ as the copolar incidence varieties of S :=
{F1 =· · ·=Fp = 0} associated with the indices 1≤i≤n−p and σ ∈Sym (n) .
The notion of a copolar incidence variety is inspired by the Room-Kempf desingularization of determinantal varieties [32, 36].
3.2 Copolar varieties
Let notations and assumptions be as in previous section and let b ∈ A(n−i)×n be a full rank matrix. We observe that the set
Veb(S) :={x∈S | ∃(λ, ϑ)∈Ap×p×A(n−p)×p : rkϑ=p and (x, b, λ, ϑ)∈Hi} does not depend on the choice of the generators F1, . . . , Fp of the vanishing ideal of S. We call the the Zariski closure in An of Veb(S) the copolar variety of S associated with the matrix b and we denote it by Vb(S) . From the argumentation at the end of the proof of Lemma 1 we deduce Veb(S) = Vb(S)∩Sreg .
Observe that a point x of S belongs to Veb(S) if and only if there exist p rows of the ((n−i)×n) –matrix b which generate the same affine linear space as the rows of the Jacobian J(F) at x. In case p := 1 and F := F1 the copolar variety Vb({F = 0}) coincides with the ith classic polar variety WKn−1−i(b)({F = 0}) of the complex hypersurface {F = 0} (here b denotes the ((n −i)×(n+ 1)) –matrix whose column number zero is a null-vector, whereas the columns number 1, . . . , n are the corresponding columns of b).
Proposition 4
If b∈A(n−i)×n is a generic matrix, then the copolar variety Vb(S) is empty or an equidi- mensional closed subvariety of dimension n−(i+ 1)p≥0 which is smooth at any point of Vb(S)∩Sreg .
Proof
(Sketch) We consider in the ambient space Fei :=An×A(n−i)×n×A(n−i)×p the Q–definable locally closed incidence variety
Hei :={(x, b, ϑ)∈eFi |x∈S, rkb =n−i, rkϑ=p, J(F)(x)T +bTϑ= 0}
Using the same argument as in Proposition 2 we see that Hei is nonempty, equidimen- sional of dimension n(n −i+ 1)−(i+ 1)p ≥ 0 and smooth. Let π : Hei 7→ A(n−i)×n be the morphism induced by the canonical projection of Fei onto A(n−i)×n. Notice that for any full rank matrix b ∈ A(n−i)×n the π–fiber of b is isomorphic to Vb(S)∩Sreg as algebraic variety. Suppose now that π is dominating. From Sard’s Theorem (see [16, 39]) we deduce that for a generic b ∈ A(n−i)×n the π–fiber of b and hence Vb(S)∩Sreg , are nonempty, equidimensional of dimension n−(i+ 1)p ≥ 0 and smooth. If π is not dominating, then we see by the same argument that Vb(S) is empty. 2 Observe that for a generic b ∈A(n−i)×n the emptiness or non-emptiness and in the latter case also the geometric degree of the copolar variety Vb(S) is an invariant of the variety S.
The incidence varieties Hi and Hi,σ may be interpreted as suitable algebraic families of copolar varieties. In [8] we considered in the case p := 1 three analogous incidence varieties which turned out to be algebraic families of dual polar varieties. Here we have a similar situation since in the hypersurface case, namely in the case p:= 1 , the copolar varieties are classic polar varieties.
4 Bipolar varieties
4.1 Definition and basic properties of bipolar varieties
In order to measure the complexity of the real point finding procedures of this paper for complete intersection varieties, we consider for 1≤p≤n, 1≤i≤n−p and σ ∈Sym (n) the generic dual polar varieties of the copolar varieties Hi and Hi,σ. In analogy to the hypersurface case tackled in [8], we call them the large and the small bipolar varieties of S.
Definition 5
The bipolar varieties B(i,j) and B(i,σ,j) are defined as follows:
• for 1≤j ≤n(n−i+ 1) +p(p−i−1) let B(i,j) a (n(n−i+ 1) +p(p−i−1)−j+ 1)th generic dual polar variety of Hi and,
• for 1 ≤ j ≤ n−p and σ ∈ Sym (n) let B(i,σ,j) a (n−p−j + 1)th generic dual polar variety of Hi,σ.
We call B(i,j) the large and B(i,σ,j) the small bipolar variety of S, respectively.
The bipolar varieties B(i,j) and B(i,σ,j)are well defined geometric objects which depend on the equation system F1(X) =· · ·Fp(X) = 0 , although the incidence varieties Hi and Hi,σ are not closed (compare the definition of the notion of polar variety in Section 2, where we have taken care of this situation). Moreover, our notation is justified because we are only interested in invariants like the dimension and the degree of our bipolar varieties and these are independent of the particular (generic) choice of the linear projective varieties used to define the bipolar varieties.
From Propositions 2 and 3 and [5], Corollary 2 we deduce that B(i,j) and B(i,σ,j) are empty or equidimensional of dimension j −1 and Cohen-Macaulay and normal at any point of B(i,j)∩Hi and B(i,σ,j)∩Hi,σ.
Let 1≤j ≤n(n−i+ 1) +p(p−i−1), a0 ∈Aj with a0 6= 0, a∗ ∈Aj×(n(n−i+1)+p(n+p−1))
generic and a := [aT0, a∗] . Furthermore, let Ta(i,j) be the polynomial (((n+ 1)p+j)× (n(n−i+ 1) +p(n+p−i))) –matrix whose first (n+ 1)p rows constitute the Jacobian of the system (1) of Section 3 and whose remaining j rows are built as in Section 2 in order to define the (n(n−i+ 1) +p(p−i−1)−j + 1) th dual polar variety of Hi associated with the linear variety K(a) . Then B(i,j) is the Zariski closure in Ti of the set of all points (x, b, λ, ϑ)∈Hi where Ta(i,j)(x, b, λ, ϑ) has not full rank.
By Tea(i,j) we denote the polynomial (((n+1)p+j−1)×(n(n−i+1)+p(n+p−i))) –matrix consisting of all rows of Ta(i,j) except the last one.
Observe that the large bipolar varieties of S form a chain of equidimensional varieties Hi %B(i,n(n−i+1)+p(p−i−1)) ⊃ · · · ⊃B(i,1).
The variety B(i,1) is empty or zero–dimensional. If B(i,1) is nonempty, then the chain is strictly decreasing. We define B(i,0) :=∅.
For t∈Nj−1 with t:= (t1, . . . , tj−1) and
(n+ 1)p < t1 <· · ·< tj−1 ≤n(n−i+ 1) +p(n+p−i)
we denote by m(i,j;t) the ((n+ 1)p+j−1) –minor of Tea(i,j) which corresponds to the first (n+ 1)p columns and the columns number t1, . . . , tj−1 of Tea(i,j). Moreover, for
(n+ 1)p < k1 <· · ·< kn(n−i+1)+p(p−i−1)−j+1≤n(n−i+ 1) +p(n+p−i)
different from t1, . . . , tj−1 and 1 ≤h≤n(n−i+ 1) +p(p−i−1)−j+ 1 we denote by Mh(i,j;t) the ((n+ 1)p+j) –minor of Ta(i,j) which corresponds to the first (n+ 1)p columns of Ta(i,j) and the columns number t1, . . . , tj−1 and kh.
Finally, for t0 ∈Nn−i and t00∈Np with t0 := (t01, . . . , t0n−i) , t00 := (t001, . . . , t00p) and 1≤t01 <· · ·< t0n−i ≤n and 1≤t001 <· · ·< t00p ≤n−i
let B(i,t0) and Θ(i,t00) be the (n−i) – and p–minors of the matrices B and Θ which correspond to the columns t01, . . . , t0n−i and rows t00, . . . , t00 of B and Θ , respectively. By induction on n(n−i+ 1) +p(p−i−1, . . . ,1) one sees easily that for any point (x, b, λ, ϑ) of B(i,j)∩Hi\B(i,j−1) there exist suitable vectors t∈Nj−1, t0 ∈Nn−i and t00 ∈Np with m(i,j;t)B(i,t0)Θ(i,t00)(x, b, λ, ϑ)6= 0.
Now Proposition 2 and Propositions 6 and 8 of [3, 4] imply that the equations of the system (1) and the equations
M1(i,j;t)= 0, . . . , Mn(n−i+1)+p(p−i−1)−j+1)(i,j;t) = 0
intersect transversally at (x, b, λ, ϑ) . In particular, the corresponding polynomials form a regular sequence in
Q[X, B,Λ,Θ]m(i,j;t)B(i,t0)Θ(i,t00)
and they define the large bipolar variety B(i,j) outside of the locus given by m(i,j;t)B(i,t0)Θ(i,t00) = 0.
Finally, observe that there exist n(n−i+1)+p(p−i−1) j−1
, n−in
and n−ip
possible choices of the vectors t ∈ Nj−1, t0 ∈ Nn−i and t00 ∈ Np, respectively. This yields
n(n−i+1)+p(p−i−1) j−1
n
n−i
n−i p
possible choices of vectors (t, t0, t00)∈Nj−1×Nn−i×Np. This considerations entail the following statement.
Proposition 6
Let notations be as above and let t ∈ Nj−i, t0 ∈ Nn−i and t00 ∈ Np be suitable vectors. Further, let D(i,j;t,t0,t00) be the closed variety of Ti defined by the condition m(i,j;t)B(i,t0)Θ(i,t00)= 0. Then the equations of the system (1) and the equations
M1(i,j;t)= 0, . . . , Mn(n−i+1)+p(p−i−1)−j+1)(i,j;t) = 0
intersect transversally at any of their common solutions in Ti \D(i,j;t,t0,t00). They define B(i,j)\D(i,j;t,t0,t00) in Ti \D(i,j;t,t0,t00). Moreover, for suitably chosen vectors (t, t0, t00) ∈ Nj−1×Nn−i×Np the union of the sets Ti\D(i,j;t,t0,t00) covers B(i,j)∩Hi\B(i,j−1). There exist n(n−i+1)+p(p−i−1)
j−1
n
n−i
n−i p
such choices for the vector (t, t0, t00)∈Nj−1×Nn−i×Np. Now let 1≤j ≤n−p, a0 ∈Aj with a0 6= 0, a∗ ∈Aj×(n(p+1)) generic and a := [aT0, a∗] . Let σ ∈Sym (n) . For the sake of simplicity of exposition we suppose that σ is the identity permutation. Furthermore, let Ta(i,σ,j) be the polynomial (((n+1)p+j)×n(p+1)) –matrix whose first (n+ 1)p rows constitute the Jacobian of the system (2) of Section 3 and whose
remaining j rows are built as in Section 2 in order to define the (n−p−j + 1) th dual polar variety of Hi,σ associated with the linear space K(a) . Then B(i,σ,j) is the Zariski closure in Fi of the set of all points (x, b, λ, ϑ)∈Hi,σ where Ta(i,σ,j)(x, b, λ, ϑ) has not full rank.
By Tea(i,σ,j) we denote the polynomial (((n+ 1)p+j−1)×n(p+ 1)) –matrix consisting of all rows of Ta(i,σ,j) except the last one.
Observe again, that the small bipolar varieties B(i,σ,j) of S form a chain of equidimensional varieties
Hi,σ %B(i,σ,n−p) ⊃ · · · ⊃ B(i,σ,1).
The variety B(i,σ,1) is empty or zero–dimensional. If B(i,σ,1) is nonempty, then the chain is strictly decreasing. We define B(i,σ,0) :=∅.
For t∈Nj−1 with t:= (t1, . . . , tj−1) and
(n+ 1)p < t1 <· · ·< tj−1 ≤n(p+ 1)
we denote by m(i,σ,j;t) the ((n+ 1)p+j −1) –minor of Tea(i,σ,j) which corresponds to the first (n+ 1)p columns and the columns number t1, . . . , tj−1 of Tea(i,σ,j). Moreover, for
(n+ 1)p < k1 <· · ·< kn−p−j+1 ≤n(p+ 1)
different from t1, . . . , tj−1 and 1≤ h ≤ n−p−j + 1 we denote by Mh(i,σ,j;t) the ((n+ 1)p+j) –minor of Ta(i,σ,j) which corresponds to the first (n+ 1)p columns of Ta(i,σ,j) and the columns number t1, . . . , tj−1 and kh.
Observe that there exist n−pj−1
choices of vectors t ∈Nj−1.
In the same way as in case of Proposition 6, now one proves the following statement.
Proposition 7
Let notations be as before and let t ∈Nj−1 be a suitable vector. Denote by D(i,σ,j;t) the closed subvariety of Fi defined by the equation m(i,σ,j;t) = 0. Then the equations of the system (2) and the equations
M1(i,σ,j;t)= 0, . . . , Mn−p−j+1(i,σ,j;t) = 0
intersect transversally at any of their common solutions in Fi \D(i,σ,j;t). They define B(i,σ,j)\D(i,σ,j;t) in Fi\D(i,σ,j;t). Moreover, for suitably chosen vectors t∈Nj−1 the union of the sets Fi\D(i,σ,j;t) covers B(i,σ,j)∩Hi,σ\ B(i,σ,j−1). There exist n−pj−1
possible choices of vectors t ∈Nj−1.
4.2 Degrees of bipolar varieties
We denote by degB(i,j) and degB(i,σ,j) the geometric degrees of the respective bipolar varieties in their ambient spaces Ti and Fi (see [26] for a definition and properties of the geometric degree of a subvariety of an affine space).
Observe that degB(i,j) remains invariant under linear transformations of the coordinates X1, . . . , Xn by unitary complex matrices.
From [8], Lemma 1 and [5], Theorem 13 we deduce that for 1≤j ≤n−p
(3) degB(i,σ,j) ≤degB(i,n(n−i))+p(p−i)+j)
holds.
Suppose that S contains a regular real point x. The there exists a permutation σ ∈ Sym (n) with ∆σ(x) 6= 0 . From Proposition 3 we deduce that (Hi,σ)R is nonempty.
This implies that Hi,σ is given by a reduced regular sequence of polynomials, namely the polynomials in the equation system (2). Moreover, the the real variety (Hi,σ)R is smooth.
Therefore we may apply [3, 4], Proposition 2 to conclude that B(i,σ,j)R contains for each connected of (Hi,σ)R at least one point. This implies
1≤degB(i,σ,1) ≤degB(i,n(n−i))+p(p−i)+1).
For 1 ≤ r ≤ p, 1 ≤ l ≤ n and σ ∈ Sym (n) we are going to analyze in the following closed subvarieties S(r,l)(i) and S(r,l)(i,σ) of the affine subspaces Ti and Fi, respectively. For this purpose we consider the lexicographical order < of the set of all pairs (r, l) with 1≤r≤p, 1≤l≤n.
Let S(r,l)(i) be the Zariski closure of the locally closed subset of Ti defined by the conditions F1(X) =· · ·=Fp(X) = 0
X
1≤s≤p
Λr0,s
∂Fs
∂Xl0 + X
1≤k≤n−i
Bk,l0Θk,r = 0, 1≤r0 ≤p, 1≤l0 ≤n, (r0, l0)≤(r, l) and rkB =n−i, rk Θ = p and rkJ(F) =p.
(4)
Observe that the particular structure of the Jacobian of the equations of system (4) implies that the corresponding polynomials form a reduced regular sequence at any of their common zeros outside of the closed locus given by the conditions
rkB < n−i, rk Θ< p or rkJ(F)< p.
Furthermore, let S(r,l)(i,σ) be the locally closed subset of Fi defined by the conditions F1(X) = · · ·=Fp(X) = 0,
X
1≤s≤p
Λr0,s ∂Fs
∂Xl0 +δr0,l0 = 0, 1≤r0 ≤r, 1≤l0 ≤p, (r0, l0)≤(r, l), X
1≤s≤p
Λr0,s∂Fs
∂Xl0
+ Θl0,r0 = 0, 1≤r0 ≤r, p < l0 ≤n−i, (r0, l0)≤(r, l), X
1≤s≤p
Λr0,s ∂Fs
∂Xl0 +Br0,l0 = 0, 1≤r0 ≤r, n−i < l≤n, (r0, l0)≤(r, l) and ∆σ(X)6= 0.
(5)
Again the particular structure of the Jacobian of the equations of system (5) implies that the corresponding polynomials form a reduced regular sequence at any of their common zeros outside of the closed locus given by the condition ∆σ(X) = 0 .
In conclusion, the polynomials of the systems (1) and (2) form strongly reduced regular sequences at any of their common zeros outside of the corresponding closed loci.
For the next statement recall that the degree of the polynomials F1, . . . , Fp is bounded by d (see Section 2).
Proposition 8
Let 1≤r≤p and 1≤l ≤n. Then we have the extrinsic estimate degS(r,l)(i) ≤dp(n+1) =dO(n2).
Proof
Without loss of generality we may suppose d ≥ 2 . Then we deduce from the B´ezout Inequality ([26, 18, 44]) that the closed subvariety of Ti defined by the equations of the system (4) is of degree at most dp(n+1) =dO(n2). 2 In fact this bound is too coarse, because refined methods, based on the multi-homogeneous B´ezout Inequality of [33], yield an estimate degS(r,l)(i) = (nnd)O(n) which is sharper for d ≥ n. This improvement will not be very relevant in Section 5 where the degree of S(r,l)(i) plays a key role in complexity estimates. More important will be the estimate degS(r,l)(i,σ) = (nd)O(n), σ ∈Sym (n) , we are going to derive now.
Lemma 9
Let 1≤ r ≤p, Λ(r) := [Λr0,s]1≤r0≤r 1≤s≤p
and ∆ := det[∂X∂Fs
l0]1≤s,l0≤p. Then the Zariski closure of the locally closed subvariety Sr of An×Ar×p defined by the conditions
F1(X) = · · ·=Fp(X) = 0, X
1≤s≤p
Λr0,s∂Fs
∂Xl0 +δr0,l0 = 0, 1≤r0 ≤r, 1≤l0 ≤p,
∆6= 0 (6)
is empty or equidimensional of dimension n−p. Its geometric degree is bounded by (nd)n. The polynomials of the system (6) form a reduced regular sequence in Q[X,Λ(r)]∆. Proof
From the equations (6) one deduces easily that for a point (x, λ) ∈ Sr with ∆(x) 6= 0 the matrix [∂X∂Fs
l0]1≤s≤r
1≤l0≤p has maximal rank. This implies that the Jacobian of (6) has full
rank at (x, λ) . Hence the variety Sr is smooth and of dimension n−p at (x, λ) . Thus Sr is empty or equidimensional of dimension n−p. Moreover, the polynomials of the system (6) form a reduced regular sequence in Q[X,Λ(r)]∆.
Observe that for x∈S with ∆(x)6= 0 there exists exactly one point λ∈Ar×p such that (x, λ) belongs to Sr. Thus Sr∩(S∆×Ar×p) is the graph of a rational map ϕ :S →Ar×p
which is everywhere defined on S∆. By Cramer’s rule each component of this rational map may be described by a rational expression whose numerator is a polynomial of Q[X]
of degree at most pd and whose denominator is ∆ .
Let K1, . . . , Kn−r be generic affine linear polynomials of Q[X,Λ(r)] . Then we have degSr = #(Sr∩ {K1 = 0, . . . , Kn−r = 0}) , where # denotes the cardinality of the corre- sponding set. Without loss of cardinality we may suppose that Sr∩ {K1 = 0, . . . , Kn−r = 0} is contained in An∆×Ar×n (see [26], Remark 2). Replacing in K1 = 0, . . . , Kn−p = 0 each indeterminate Λr0,s, 1≤r0 ≤r, 1≤ s≤ p by the given rational expression for the corresponding coordinate of ϕ and cleaning the denominator ∆ we obtain together with F1, . . . , Fp a system of n polynomials of Q[X] having degree at most pd which in An∆
defines a set of the same cardinality as Sr∩ {K1 = 0, . . . , Kn−r = 0}. From the B´ezout Inequality we deduce therefore
degSr ≤(pd)n ≤(nd)n.
2 Proposition 10
Let 1≤r≤p and 1≤l ≤n. Then we have the estimate degS(r,l)(i,σ)≤(nd3)n. Proof
Without loss of generality we may suppose that σ ∈Sym (n) is the identity permutation.
Then we have with the notation of the previous lemma ∆ = ∆σ.
We consider Sr−1×Ap and Sr as closed subvarieties of An×Ar×p with the convention S0 :=S∆. Let V(r,l) be the Zariski closure of the locally closed subvariety of An×Ar×p defined by the conditions
F1(X) = · · ·=Fp(X) = 0, X
1≤s≤p
Λr0,s ∂Fs
∂Xl0 +δr0,l0 = 0, 1≤r0 ≤r, 1≤l0 ≤p, (r0, l0)≤(r, l) and ∆σ 6= 0
Observe that V(r,l) is the intersection of Sr−1 ×Ap with the subvariety of An ×Ar×p defined by the equations
X
1≤s≤p
Λr,s∂Fs
∂Xl0 +δr,l0 = 0, 1≤l0 ≤min{l, p}.
From the B´ezout Inequality and Lemma 9 we infer
degV(r,l) ≤dp degSr ≤(n d2)n.
